Is a multilayer perceptron a recursive function?
Inherently, no. The MLP is just a data structure. It represents a function, but a standard MLP is just representing an input-output mapping, and there's no recursive structure to it.

On the other hand, possibly your source is referring to the common algorithms that operate over MLPs, specifically forward propagation for prediction and back propagation for training. Both of these algorithms are easy to think about recursively, with each node performing a sort of recursive call with its children or parents as the target, and some useful information about activations or errors attached. I actually encourage my students to implement it recursively for this reason, even though it's probably not the most efficient solution.